ATLANTA—June’s Supreme Court ruling on same-sex marriage is sending some couples to court for a very different reason: same-sex divorce.

Marie Griffin, who lives in Savannah, Ga., moved out of the home she shared with her wife in late May. But the couple, who married in 2014 in Washington, D.C., couldn’t legally end their marriage, because Georgia didn’t permit either same-sex marriage or divorce.

On the day of the June 26 Supreme Court ruling that all states must recognize same-sex marriages, she said that she recalled thinking, “We have to, and we can, get a divorce now.” She contacted a lawyer and is pursuing a divorce.